Theorem dalem38 36848

Description: Lemma for dath 36874. Plane 𝑌 belongs to the 3-dimensional volume 𝐺𝐻𝐼𝑐. (Contributed by NM, 5-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
dalem38.m	⊢ ∧ = (meet‘𝐾)
dalem38.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem38.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem38.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem38.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem38.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem38.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))

Assertion

Ref	Expression
dalem38	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))

Proof of Theorem dalem38

Step	Hyp	Ref	Expression
1		dalem38.y	. 2 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
2		dalem.ph	. . . . . . 7 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
3		dalem.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
4		dalem.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
5		dalem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6		dalem.ps	. . . . . . 7 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
7		dalem38.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
8		dalem38.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
9		dalem38.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
10		dalem38.g	. . . . . . 7 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
11	2, 3, 4, 5, 6, 7, 8, 1, 9, 10	dalem28 36838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐))
12		dalem38.h	. . . . . . 7 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
13	2, 3, 4, 5, 6, 7, 8, 1, 9, 12	dalem33 36843	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐))
14	2	dalemkelat 36762	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
15	14	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
16	2, 5	dalempeb 36777	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
17	16	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾))
18	2	dalemkehl 36761	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
19	18	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
20	2, 3, 4, 5, 6, 7, 8, 1, 9, 10	dalem23 36834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
21	6	dalemccea 36821	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
22	21	3ad2ant3 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
23		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
24	23, 4, 5	hlatjcl 36505	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾))
25	19, 20, 22, 24	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾))
26	2, 5	dalemqeb 36778	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
27	26	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾))
28	2, 3, 4, 5, 6, 7, 8, 1, 9, 12	dalem29 36839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
29	23, 4, 5	hlatjcl 36505	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐻 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))
30	19, 28, 22, 29	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))
31	23, 3, 4	latjlej12 17679	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝑐) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))))
32	15, 17, 25, 27, 30, 31	syl122anc 1375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))))
33	11, 13, 32	mp2and 697	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
34	23, 5	atbase 36427	. . . . . . 7 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
35	20, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
36	23, 5	atbase 36427	. . . . . . 7 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
37	28, 36	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
38	6, 5	dalemcceb 36827	. . . . . . 7 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
39	38	3ad2ant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
40	23, 4	latjjdir 17716	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
41	15, 35, 37, 39, 40	syl13anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
42	33, 41	breqtrrd 5096	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐))
43		dalem38.i	. . . . 5 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
44	2, 3, 4, 5, 6, 7, 8, 1, 9, 43	dalem37 36847	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))
45	2, 4, 5	dalempjqeb 36783	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
46	45	3ad2ant1 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	23, 4, 5	hlatjcl 36505	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
48	19, 20, 28, 47	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
49	23, 4	latjcl 17663	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾))
50	15, 48, 39, 49	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾))
51	2, 5	dalemreb 36779	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
52	51	3ad2ant1 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
53	2, 3, 4, 5, 6, 7, 8, 1, 9, 43	dalem34 36844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
54	23, 4, 5	hlatjcl 36505	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))
55	19, 53, 22, 54	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))
56	23, 3, 4	latjlej12 17679	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∧ 𝑅 ≤ (𝐼 ∨ 𝑐)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))))
57	15, 46, 50, 52, 55, 56	syl122anc 1375	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∧ 𝑅 ≤ (𝐼 ∨ 𝑐)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))))
58	42, 44, 57	mp2and 697	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
59	23, 5	atbase 36427	. . . . 5 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
60	53, 59	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
61	23, 4	latjjdir 17716	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
62	15, 48, 60, 39, 61	syl13anc 1368	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
63	58, 62	breqtrrd 5096	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
64	1, 63	eqbrtrid 5103	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  joincjn 17556  meetcmee 17557  Latclat 17657  Atomscatm 36401  HLchlt 36488  LPlanesclpl 36630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-lat 17658  df-clat 17720  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489  df-llines 36636  df-lplanes 36637

This theorem is referenced by:  dalem39  36849